Generally, a damping unit of an offset printing press as disclosed in DE-PS 3 623 590 includes a damping fluid ductor roller which transfers damping fluid from a fountain or trough to an inking roller and a metering roller which engages the fountain or ductor roller to regulate the thickness of the damping fluid. As is the normal practice, the ductor roller and the metering roller are driven independently of the press by a separate variable-speed geared motor by way of a clutch, a first intermediate gear and a driven gear, the driven gear being securely disposed on the journal of the damping fluid ductor roller. Another intermediate gear provides an automatic drive of the metering roller by way of a second driven gear either synchronously, without peripheral slip, or, with an altered transmission ratio, asynchronously, with slip, with respect to the ductor roller. When the damping and inking unit are washed together in a conventional manner as disclosed for example, from DE-OS 3 641 013, the drive with slip presents a problem because the damping fluid fountain roller and the metering roller run with peripheral slip unless disengaged from one another, so that since they run at different circumferential speeds the rollers could be damaged during washing. If the metering roller is disengaged from the damping unit fountain roller, the metering roller then has to be washed manually.